Vertex-pancyclicity of the (n,k)-bubble-sort networks

Abstract

The cycle embedding is an important problem of networks, which can determine the fault tolerance of the networks. A network can be viewed as a graph. Let m be an integer with m≥4, G be a graph and w∈V(G) be an arbitrary vertex. The graph G is vertex-pancyclic if G has a cycle Cl of length l with w∈V(Cl) for every l∈{3,4,⋯,|V(G)|} and G is m-weak-vertex-pancyclic if G has a cycle Cl of length l with w∈V(Cl) for every l∈{m,m+1,⋯,|V(G)|}. Let G′ be a bipartite graph and w′∈V(G′) be an arbitrary vertex. The graph G′ is vertex-bipancyclic if G′ has a cycle Ch of length h with w′∈V(Ch) for any even integer h with 4≤h≤|V(G′)|. In this paper, we study the cycle embedding in the (n,k)-bubble-sort network Bn,k. We obtain that (1) Bn,1 is vertex-pancyclic for n≥3. (2) Bn,n−1 is vertex-bipancyclic for n≥4. (3) B4,2 and B5,2 are 6-weak-vertex-pancyclic and B5,3 is vertex-pancyclic. (4) Bn,k is vertex-pancyclic for n≥6 with 2≤k≤n−2 and every constructed cycle of Bn,k contains a residual edge for n≥4 with 2≤k≤n−2.